1996 3: 243-256Learn. Mem. O Jensen, M A Idiart and J E Lisman channels.networks with theta/gamma oscillations: role of fast NMDA Physiologically realistic formation of autoassociative memory in  

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Physiologically Realistic Formation of Autoassociative Memory in Networks with Theta/Gamma Oscillations: Role of Fast NMDA Channels Ole Jensen, ~ Marco A.P. Idiart, 2 and John E. Lismanl'3 lV01en Center for Complex Systems Brandeis University

Waltham, Massachusetts 02254 2Institut0 Fisica

UFRGS

91501-970 Porto Alegre, RS Brazil

Abstract

Recordings f rom brain regions involved in m e m o r y funct ion show dual oscillations in which each cycle of a low-frequency theta oscillation (5-8 Hz) is subdivided into about seven subcycles by h igh f requency g a m m a oscillations (20-60 Hz). It has been p roposed (Lisman and Idiart 1995) that such networks are a mul t ip lexed shor t - term m e m o r y (STM) buffer that can actively main ta in about seven memor ies , a capability of h u m a n STM. A m e m o r y is encoded by a subset of principal neu rons that fire synchronous ly in a part icular gamma subcycle. Firing is main ta ined by a m e m b r a n e process intrinsic to each cell. We now extend this mode l by incorporat ing recur ren t connect ions with modifiable synapses to store long- te rm m e m o r y (LTM). The repet i t ion provided by STM gradually modif ies synapses in a physiological ly realistic way. Because different memor i e s are active in different gamma subcycles, the format ion of autoassociative LTM requires that synaptic modif icat ion depend on N-methyl-D-aspartate (NMDA) channels having a t ime constant of deactivation that is of the same order as the dura t ion of a gamma subcycle (15-50 msec). Many types of NMDA channels have longer t ime constants (150 msec), as for instance those found in the h ippocampus , but both fast and slow NMDA channels are presen t in

3Corresponding author.

cortex. This is the first proposal for the special role of these fast NMDA channels . The STM for novel i tems mus t d e p e n d on activity-dependent changes intr insic to neu rons ra ther than r ecu r ren t connect ions , which have not developed the r equ i red selectivity. Because these intrinsic mechan i sms are not error-correct ing, STM will become slowly cor rup ted by noise. This limits the accuracy with which LTM can become encoded after a single presenta t ion. Accurate encoding of i tems in LTM can be achieved by mul t ip le presentat ions , provided different m e m o r y i tems are presen ted in a varied inter leaved order. Our results indicate that a l imited memory-capaci ty STM m o d e l can be integrated in the same ne twork with a high-capacity LTM model .

Introduct ion

Insights into the mechanisms of memory have come from several different fields. Psychophysical work indicates the existence of a short-term mem- ory (STM) mechanism of limited capacity (7---2 items) (Miller 1956) and a long-term memory (LTM) mechanism of much higher capacity. Phys- iological work indicates that STM is maintained by persistent cell firing (Fuster and Jervey 1982; Goldmann-Rakic 1995), whereas LTM appears to be maintained by modifications of synaptic strength (Bliss and Collingridge 1993; Morris et al. 1986). Theoretical analysis of neural networks has indicated how recurrent collaterals with modifi- able synapses can lead to distributed storage of

LEARNING & MEMORY 3:243-256 © 1996 by Cold Spring Harbor Laboratory Press ISSN1072-0502/96 $5.00

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LTM (Hopfield 1982; Amit 1989; Hertz et al. 1991). Most recently it has been found that net- works involved in memory storage show complex oscillatory dynamics. The hippocampus and some regions of cortex have dual oscillations in which a low-frequency oscillation in the theta frequency range (5 -8 Hz) is subdivided into subcycles by a high-frequency oscillation in the gamma (20-60 Hz) range (Biedenbach 1966; Bressler 1984; Solt- esz and Deschenes 1993; Bragin et al. 1995).

We have been interested in synthesizing the results from these fields. Our initial work focused on the relationship between network oscillations and the psychophysical properties of STM (Lisman and Idiart 1995). We proposed that dual oscilla- tions serve as a multiplexing mechanism by which multiple short-term memories can be actively maintained in the same network. A STM is assumed to be encoded by a subset of cells that fire syn- chronously during a given gamma subcycle, an as- sumption consistent with previous work indicat- ing the importance of synchrony (Gray et al. 1989; Abeles et al. 1993). Because there are about seven gamma cycles in a theta cycle, this could explain why STM is limited to seven items. Further sup- port for this view is that tb~ Sternberg number (Sternberg 1966), which is a psychophysical mea- surement of the temporal separation of memories, approximately equals the period of a gamma oscil- lation. The proposition that different information is stored at different phases of a theta cycle (i.e., in different gamma subcycles) is made plausible by studies of hippocampal place cells (O'Keefe and Recce 1993; Skaggs et al. 1996) showing that dif- ferent cells fire with different phases during the theta cycle.

A second feature of our previous model is the mechanism by which the firing that underlies STM is maintained. Action potentials triggered at the time of information input to the network trigger an after-depolarization (ADP) that is intrinsic to each neuron (Lisman and Idiart 1995). An ADP of this kind has been observed in hippocampal and cortical neurons (Andrade 1991; Caeser et al. 1993; Storm 1989; Libri et al. 1994). The ADP has sufficient duration to trigger firing on subsequent theta cycles and thereby maintains STM. This mech- anism for maintaining firing stands in contrast to reverbatory mechanisms involving loops of inter- connected cells (Hebb 1949). In our previous STM model (Lisman and Idiart 1995) there was no mechanism for LTM (Fig. 1A). In contrast standard neural network models (Hopfield 1982; Amit

A B Informational Input

Osfilla[°rY~] ~1 L1 ~] F e d i d [ ] l l (gam~imC°a~ ( t h e t a ~

Informational Input Modifiable Synapses "-4-,

;~ - Recurrent ,I _- Collaterals

C Modifiable Informational Input Synapses /

(NMDA/AMPA).....t. L'

I

Oscillatory I >----- 3----- > Input - - I

(them)

- Recurrent - Collaterals

- - - ~ Feedback Inhibition (gamma)

Figure 1 : The goal of this series of papers is to examine networks that integrate the STM and LTM networks. (A) STM network in which information is stored using the ADP that is intrinsic to each pyramidal cell. Dual oscil- lations occur because of external theta-frequency input and because of feedback inhibition from an inhibitory interneuron, which generates the gamma cycles. (B) Au- toassociative network in which LTM is encoded in mod- ifiable recurrent synapses. (C) Integrated network that has both STM and LTM. Synaptic modification is gov- erned by fast NMDA channels. Synaptic transmission is mediated by AMPA channels.

1989; Hertz et al. 1991) have recurrent collaterals with modifiable synapses (Fig. 1B) for storing LTM, but no mechanism for storing novel STMs. Our goal in this and the subsequent papers in this series is to examine networks that have both STM and LTM. As shown in Figure 1 C, the ADP feature of our STM model and the modifiable synapses of standard LTM models are compatible and inte- grated models can be analyzed.

By studying the interaction of STM and LTM, we have attempted to capture temporal complex- ities of real learning situations and the kinetic properties of the cellular processes that underlie learning. It is a general assumption of our work that synaptic modification similar to long-term po- tentiation (LTP) and long-term depression (LTD) underlies the storage of LTM (Morris et al. 1986), even though this is not firmly established. Exper- imental evidence indicates that LTP occurs slowly and requires repetitive firing (Larson et al. 1986; Hanse and Gustafsson 1994). Specifically, experi- ments on hippocampal LTP show that induction

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requires a second or more high-frequency stimu- lation unde r s tandard condi t ions (Bliss and LOmo 1973). Once LTP has b e e n induced it takes addi- t ional t ime ( tens of seconds) to express itself (Hanse and Gustafsson 1994). Thus a STM buffer is needed bo th to cause the pers is tent firing requi red for transfer of informat ion to LTM and to mainta in informat ion unt i l LTM is expressed. An addit ional complex i ty is that m e m o r y i tems often arrive in s t reams of mul t ip le i tems per second rather than as single items. An STM buffer that can capture mul t ip le i tems and actively mainta in them is capa- ble of deal ing wi th such streams.

In this paper we deal specifically wi th the learning of n o v e l items. An example wou ld be the first t ime one is exposed to a digit. We show that this novel ty creates special p rob lems for STM and the cor rec t format ion of autoassociative LTM. In a different class of STM tasks, the i tems themselves are already known (e.g., the digits); what has to be r e m e m b e r e d is w h i c h i tems are present and in what order, as for instance in a p h o n e number . The fact that the individual i tems are familiar implies that they are already stored autoassociatively in LTM. In the second paper in this series (Jensen and Lisman 1996b, this issue) we show how these rep- resentat ions in LTM can dramatical ly enhance the rel iabil i ty wi th w h i c h arbitrary lists of such i tems can be s tored in STM.

A crit ical insight to emerge from our analysis of ne tworks wi th the t a /gamma oscillations regards the impor tance of the kinetics of N-methyl-D-aspar- tate (NMDA) channels . These channels control the induc t ion of LTP at Hebbian synapses (for review, see Bliss and Coll ingridge 1993). Recent work shows that different subtypes of the NMDA chan- nel have very different t ime constants of deactiva- t ion (Hes t r in et al. 1994; Monyer 1994; S. Vicini, J.F. Wang, J.S. Poling, and D.R. Grayson, pers. comm.) . Fast NMDA channels have a t ime constant on the same scale as a gamma cycle, but o ther isoforms have m u c h longer t ime constants. In this paper w e show that fast NMDA channels ('r ~ 25 m s e c ) have kinet ics that are appropriate for the format ion of autoassociative LTM. We show that NMDA channels w i th a slow t ime constant for de- activation (a- ~ 150 m s e c ) wou ld not be wel l sui ted for forming autoassociative memories , but are ideally sui ted for forming heteroassociat ive m e m o r y sequences. In the third paper in this se- ries ( Jensen and Lisman 1996c, this issue), we show h o w networks wi th slow NMDA channels can form LTM for sequences of known i tems (e.g.,

a phone number ) . We show that learning se- quences of known i tems can occur after a single presentation. Therefore, this is a mode l of episodic memory. Because h ippocampal synapses have only slow NMDA channels (Forsythe and Wes tbrook 1988; Jahr and Stevens 1990), an impl ica t ion is that the h ippocampal CA3 region, long thought to be an autoassociative network, may actually be a heteroassociat ive ne twork special ized for se- quence learning. The ability of such heteroassocia- tive models to quanti ta t ively account for impor- tant proper t ies of h ippocampal p lace cells is de- scr ibed in a final paper in this series ( Jensen and Lisman 1996a, this issue).

Materials and Methods

The field of neuronal mode l ing spans from im- p lementa t ion of abstract b inary neurons to de- tailed compar tmenta l models. An in te rmedia te ap- proach has been used here. The neurons are sim- plified to have a single compar tment , bu t impor tant temporal character is t ics of some mem- brane processes are expl ic i t ly modeled . We have i m p l e m e n t e d the ne twork using spiking mode l neurons so that STM can be e n c o d e d by synchrony of spikes rather than by rates (Gers tner et al. 1993; Hopfield and Hertz 1995).

The archi tecture of the ne twork is i l lustrated in Figure 1C. The net cur ren t to each pyramidal neuron in the ne twork is approx imated by

inet = ileak + iADP + iA + itheta + iGABA + isyn + iext (1)

For s implici ty w e have ignored the proper t ies of the m e m b r a n e capacitance. The main effect of the capaci tance is to slow down transients and to filter noise. Because we have not m o d e l e d the mem- brane capaci tance expl ic i t ly w e have slightly s lowed the kinetics of synaptic currents , so that the result ing synaptic voltage events have reason- able kinetics.

V( t) = Rmembrane( iADp( t ) + iA ( t) + itheta( t) + iGABA ( t)

+ isyn(t) + iexAt)) + Wrest ( 2 )

The average input resis tance in pyramidal CA3

cells has been found to be ~Rmembrane ~- 33 Mfl (Turner and Schwartzkroin 1983).

SPIKING

We have not i m p l e m e n t e d the fast currents involved in act ion potent ials and bursting. A spike

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is m o d e l e d as an instantaneous event that occurs w h e n threshold is reached, Vthre s = - - 5 0 mY. The cell is then reset to Wrest ~- --60 mY.

AFTER-DEPOLARIZATION (ADP)

Pyramidal cells exhib i t an after-depolarization (ADP) after spiking dur ing chol inergic (Storm 1989; Andrade 1991; Caeser et al. 1993; Libri et al. 1994) or serotonergic modula t ion (Araneda and Andrade 1991 ). The ADP provides a ramp of ex- ci tat ion that bui lds up after a spike, wi th in a t ime of 200 msec and then falls. We mode l iAo P by an alpha function:

t-- tfire ( t ~ tfire'~ exp 1 (3 ) iADe(t) = AAD P rADP fADe /

W h e n e v e r the cell spikes again, the ADP is reset and starts to bui ld up again. We have used "rAp e = 200 msec. This means that the ADP starts to dec l ine just after a subsequen t theta cycle unless the ADP is refreshed. It is this decl ine that under- lies the dropout of a previously inser ted m e m o r y w h e n an e ighth m e m o r y is in t roduced (see Fig. 5 of J ensen and Lisman 1996b, this issue). A conse- quence of p icking the T of the ADP in this way is that the ADP is fairly flat (Fig. 2c) at the t ime w h e n the ADP triggers spiking. It is for this reason that the t iming ramps provided by the ADP can be dis- rup ted by low levels of noise. We suspect that a more c o m p l e x and realistic mode l of the ADP wou ld al low more realistic choices of m e m b r a n e noise.

FAST AFTER-HYPERPOLARIZATION (AHP)

The fast AHP cur ren t prevents pyramidal cells f rom fast, repet i t ive firing. We mode l the current by a decreas ing exponent ia l :

t --- tfire ~ iA(t) = AA exp q'A ] (4)

w h e r e rA = 5 msec (see Storm 1990 for approxi- mate values).

THETA OSCILLATIONS

Theta oscil lat ions have been identif ied in the h ippocampus (Vanderwol f 1969; Stewart and Fox

pattern #1 pattern #2 + ¢

C ~

• I [ l°l I ~ I I

• : 6 I I I I I I

[ ', I ', I I I I l l I 1 0 0 ' 2 1 0 ' 3010 ' 4 0 0 ' 5 1 0 ' 6 / 0 ' 7 1 0 ' 810 ' 9 / 0 ' 1000

0 1

-80 1 ' I ' I ' I ' I ' I ' ! ' I ' +!+ ' ._I 100 200 300 400 500 600 700 800 900 1000

200

100

0 100 200 300 400 500 600 700 800 900 1000

200 D g 8 1 ~ ~

..# -lOO -200

100 200 300 400 500 600 700 800 900 1000

E o

g -100 -150

.~ -200 -250 I

100 200 300 400 500 600 700 800 900 1000

time (ms)

Figure 2: Mechanism of STM network (Fig. 1A). The first memory pattern is introduced at t= 160 msec. The solid line in (B) is the membrane potential from cell 1 that encodes pattern 1. The broken line is the membrane potential from cell 6 that encodes pattern 2. The ramp of excitation, the ADP, is shown for cell 1 and cell 6 in C. The spiking of an activated cell is maintained by the ADP and the frequency of the spiking is locked to the subthreshold theta input (D). At t= 500 msec the second pattern is introduced. As seen in E, the feedback inhibi- tion keeps the memories separated in time.

1991; Bragin et al. 1995), en torh ina l cor tex (Green and Arduini 1954; Mitchel l and Ranek 1980), the cingulate cor tex (Leung and Borst 1987) and neocor tex (Biedenback 1966; Silva et al. 1991; Nakamura et al. 1992). The media l sep- tum provides both chol inergic and GABAergic project ions to the e n t o r h i n a l - h i p p o c a m p a l - e n - torhinal loop (Stewart and Fox 1990). Further- more, temporal correlat ions b e t w e e n active cells in the septum and the h ippocampal system indi- cate that not only does the sep tum provide a con- stant chol inergic modula t ion that facilitates oscil- lations, but it also induces a phasic drive (Alonso et al. 1987). Regions in media l forebrain having the same oscillatory proper t ies as the sep tum are known to project to cort ical areas (Alonso et al. 1996). We imagine that the same type of projec- tions drives certain cortical regions at theta fre-

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quency just as the sep tum drives the hippocam- pus. Hence the oscil latory input to cor tex and hip- pocampus is m o d e l e d as

itheta(t) = BthetaSin(2ffrft) ( 5 )

w h e r e the theta f requency f = 6 Hz.

SYNAPTIC INPUT

The recur ren t collaterals are all exci ta tory and make one to one connec t ions wi th all of the pyra- midal cell. The synaptic t ransmiss ion is media ted by synaptically re leased glutamate b ind ing to AMPA and NMDA receptors. The synapt ic input to

cell i is

FEEDBACK INHIBITION

In general, in te rneurons have been found to provide a powerfu l inhib i tory feedback to the py- ramidal cells, (Andersen et al. 1963; Andersen et al. 1964; Cobb et al. 1995). Fur thermore, intra- cel lular recordings, in vivo, f rom basket cells in the rat h ippocampus show that these cells fire at gamma f requency (Sik et al. 1995) on the posit ive por t ion of the theta oscillation. Place cell record- ings of rats show that in te rneurons fire at a m u c h h igher f requency than the pyramidal neurons, and that they have no spatial selectivity Clung et al. 1994). This suggests that the inhibi tory feedback could be responsib le for generat ing the gamma oscil lat ions in the fol lowing way: The firing of a subset of the pyramidal neurons will exci te the ent ire in te rneurona l ne twork through converging exci ta tory inputs. The in te rneurons wil l then pro- vide an inhib i tory feedback to all pyramidal cells. After the inhib i t ion wears off, a n e w subset of py- ramidal neurons wil l b e c o m e active and so forth. Because we assume that all in te rneurons fire in synchrony w e s imply mode l the net GABAergic input as

iGaeA(t) =

aN J ~ TGABA / e x p 1 TGABA / (6 )

w h e r e t j refers to the act ion potent ia l of the j t h pyramidal cell. We have chosen TGABA = 4 msec (Miles 1990). The te rm a N serves to normal ize the GABAergic input if the ne twork size is changed. N is the n u m b e r of pyramidal neurons in the net- work. The constant a denotes the sparseness of pyramidal cells coding for informat ion (i.e., a = M / N w h e r e M is the n u m b e r of cells represent- ing a pat tern) . In these papers we always keep M constant: M = a N = 5.

IAMPA(t) + INMDA(t) (7) iisyn(t) = .i .i

w h e r e the AMPA curren t is m o d e l e d by an alpha- funct ion

-

N ( ) AAMPA E ij t - l~ire- tdelay

aN W syn TAMPA J

( t--~ire~-tdelaY~TAMPA / exp 1 - _ (8)

and the NMDA curren t by a double exponent ia l

iiXmhA(t) =

ANMDA • O' t - - t~ire ~ tdelay w;y n exp -

aN TNMDA, f / J

(9 )

hcre ~J W~y,, is the relative ampl i tude of an EPSC in cell i evoked by cell j . For the t ime course of the AMPA-mediated EPSCs we have chosen TaMpa = 1.5 msec (Wil l iams and Johns ton 1991). In this paper (bu t not in J ensen and Lisman 1996a,c, this issue) w e assume that only AMPA receptors part icipate in p roduc ing the EPSP; ANMDA = 0 mA. However the fall and rise t ime for the NMDA c o m p o n e n t is of impor tance for

synaptic modification, TNMDAf=7.0 m s e c and TNMDa,r = 1.0 msec. The delay in the recur ren t feedback is tdelay = 0.5 msec.

SYNAPTIC MODIFICATION

The repet i t ion of m e m o r y pat terns in the short- term m e m o r y buffer allows us to cons ider

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physiological realistic learning rules. Long-term potent ia t ion (LTP) is a possible physiological im- p lementa t ion of the Hebb-rule: Simultaneous pre- and postsynapt ic activity enhances synaptic effi- cacy. The e n h a n c e m e n t has been shown to de- p e n d on NMDA recep to r s that are act ivated by the binding of glutamate. The activated NMDA recep- tors al low an inward cur ren t of Ca 2 +, provided there is sufficient postsynapt ic depolarization. Through less unde r s tood mechanisms, the Ca 2+ rise tr iggers LTP by increasing AMPA- and NMDA-media ted synaptic responses (for review, see Bliss and Collingridge 1993). We assume that the pos tsynapt ic depolar izat ion is at tr ibutable to back-propagat ing act ion potentials or o ther den- dritic depolar iz ing events that occur wi th a small delay relative to the spike initiation in the somatic region (Markram et al. 1995):

t ( t) ipost(t) *post exp 1 ~os (lO)

w h e r e "rpost = 2.0 msec is the durat ion of a back- propagat ing spike at the postsynapt ic spine.

As ment ioned , the kinetics of the NMDA chan- nels is going to be crucial for the funct ion of the ne twork. We mode l the t ime course of the gluta- mate bound to the NMDA recep tors by a double exponent ia l :

,lU,t,: xp( ( 1 1 )

We apply the values TNMDAf=7.0 m s e c and "~NMDA,r = 1.0 msec to character ize the kinetics of the fast NMDA recep to r s (Monyer et al. 1994; S. Vicini, J.F. Wang, J.S. Poling, and D.R. Grayson, pers. comm.; see Discussion). Finally, the synaptic s t rength f rom cell i to cell j is de t e rmined by:

dwiJ~n m

dt

ipost(t- t~ire)" bglu(t- tfirei _ tdelay ) (1 - O W;~n) ~pp

+ \(ip°s~t--'rnpp ~ t~ire) + bglu(t- "rpnp-- t~ire -- tdelay))

( 0 - W~yn) ( 1 2 )

w h e r e "rdetay is the t ime it takes an act ion potential

L E A R N / N G

to travel f rom the soma to the synapses of the r ecur ren t collaterals. The first t e rm in the expres- sion is responsible for implement ing Hebbian LTP.

If bglu ( t ) a n d ipost ( t ) peak at the same time, w/j will approach one wi th the character is t ic t ime "~pp (as-

s u m i n g "~pp < < "rnp p and Zpp < < Zpnp)" If the pre- and postsynapt ic events are slightly asynchronous , the potent ia t ion is g raded wi th respec t to the tem- poral separat ion of the events. If the events are fully asynchronous a decrease in synaptic efficacy occurs: LTD (for review, see Christie et al. 1994; Linden 1994). This is mode led by the second term. If a presynapt ic event occurs w i thou t a postsynapt ic event, the synaptic we igh t w# ap- proaches zero wi th a character is t ic t ime "gpnp (pre-, not post-). Similar if only a pos tsynapt ic event occurs, wij approaches zero wi th the t ime Tnpp (no t pre-, post-). We apply the values Zpp = 50 msec and "rpnp=Tnpp = 250 msec. Note that the learning rule we use causes the value at wh ich synapses saturate to be graded wi th respec t to the convolved pre- and postsynapt ic activity no mat te r h o w many times st imulat ion is repeated.

NOISE

As a computa t ional conven ience w e add Gaus- sian noise to the firing threshold of each pyramidal cell at each gamma cycle. The s tandard deviat ion is (r = 0.05 mV.

EXTERNAL INPUT

The external input iex t s e r v e s to exci te pyra- midal neurons w h e n memor ie s are in t roduced. In the simulations we always in t roduce novel pat- terns at the t roughs of theta cycles.

We have used the cons tants Aaoe = 300 pA,

Btheta = 150 pA, A A = -- 120 pA, AGAnA = -- 180 pA and AAMPA----700 pA. Slightly different pa ramete r s are used in some of the simulations in Jensen and Lisman (1996a,b ,c this issue). See figure capt ions for details. When calculating the m e m b r a n e poten- tials numerical ly w e use the t ime step A t = 0.1 msec. In the simulations w e use a n e t w o r k of 10 to 40 cells. We use relatively few cells in our simu- lations for the sake of fast simulations.

R e s u l t s

GENERAL PROPERTIES OF NETWORK FUNCTION

Before p roceed ing wi th issues re la ted to LTM, it will be useful to descr ibe h o w the STM n e t w o r k

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acts as a buffer to capture mul t ip le i tems as they occur in real time. An STM is represen ted by the co inc iden t firing of a subset of neurons wi th in a par t icular gamma cycle. Different memor ie s are e n c o d e d by different subsets. In Figure 2A,B one m e m o r y is inser ted into the buffer at 160 msec by an external input that synchronous ly fires five of the cells shown. This firing triggers a slowly rising ADP. The ADP causes the cells to fire on the sub- sequent theta cycle (Fig. 2A,B). W h e n this hap- pens, the ADP ramp is reset ( re f reshed) making it possible for the same processes to occur on the nex t theta cycle. A second m e m o r y is inserted 340 msec after the first in the t rough of the theta cycle. This leads to synchronous firing of two other cells. On subsequen t theta cycles, the dynamics of the ne twork lead to the firing of this m e m o r y in the second gamma cycle. Gamma cycles are separated by 12 msec (80 Hz) in this simulation. The fact that the memor i e s occur red 340 msec apart, but are recrea ted wi th a separat ion of 12 msec, is what is t e rmed t ime compress ion (Tank and Hopfield 1987; Skaggs et al. 1996). Because mult iple mem- ories are actively main ta ined in different t ime slots, we refer to this as a mul t ip lexed STM buffer.

The funct ion of the ne twork can be under- stood in terms of the c o m p o n e n t currents of the pyramidal cells (Fig. 2C-E). The ADP causes per- sistent firing and controls the t iming of the firing. In the r ight part of Figure 2C it can be seen that the ramps for the two memor ie s are slightly offset in t ime and it is this offset w h i c h brings cells that encode the first m e m o r y to threshold before cells that encode the second (wi th in each theta cycle).

Figure 2E shows the feedback inh ib i t ion that fol- lows each act ion potential . It is this inhib i t ion that serves to restrict firing to discrete phases of the theta oscillation.

The order in w h i c h a set of memor i e s is re- peated in the STM buffer is i n d e p e n d e n t of the exact phase of in t roduc t ion of n e w memor ies . Fig- ure 3 shows what happens w h e n a four th m e m o r y is p resen ted to a ne twork that already contains three active memories . As can be seen, the fourth m e m o r y is always stored in the four th gamma cy- cle. Over a cer tain range, the exact phase of intro- duct ion is not impor tant (phases dur ing active fir- ing of memor ie s already in the buffer were not examined) . It wou ld seem to be impor tan t that a n e w m e m o r y is not in t roduced dur ing the per iod that previously inser ted memor i e s are active and we imagine that a ne twork earl ier in the sensory pathway has the capabil i ty of storing informat ion for up to several h u n d r e d mi l l i seconds and con- troll ing the entry of informat ion into the short- term buffer. The discret izat ion of sensory process- ing by oscillations, even at the level of the ret ina (Neuenschwande r and Singer 1996) and the very short-term m e m o r y processes t e rmed iconic (Long 1980) and echoic m e m o r y presen t in pri- mary sensory cor tex (Wicke lgren 1969; Lu et al. 1992) may fulfill these requi rements .

LIFETIME OF NMDA CHANNELS COMPARED WITH THE DURATION OF A GAMMA CYCLE

We now consider h o w LTM format ion wou ld occur if the STM buffer descr ibed above had re-

A B

500 600 700 800 900 1000 500 600 700 800 900 1000

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Figure 3" The order of memories in the STM buffer is independent of the exact phase at which they are introduced. A new memory (#4) is introduced to the network at different phases. As seen, the new memory always ends up being the last to be activated in the buffer (i.e., the order of the memory items in the STM is inde- pendent of when they are presented). The new memory is introduced at the phases 0, 270, 90, and 170. Introduction of new memory during the activity of an old memory was not consid- ered (see text).

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cur ren t connec t ions wi th modif iable synapses. We assume that all pr incipal neurons are intercon- nec ted by recurrents (Fig. 1C). Physiological work on Hebb ian synapses allows us to predic t what will happen at these recur ren t synapses. Although as stated by Hebb, synchronous firing in the presyn- aptic and postsynapt ic cell is the condi t ion for s t rengthening synapses, the exact mean ing of syn- ch rony was not defined. Exper iments show that LTP can occur even w h e n the postsynaptic depo- larization occurs wi th a delay after presynapt ic fir- ing (Levy and Steward 1983; Gustafsson and Wig- strOm 1986; Gustafsson et al. 1987; Larson and Lynch 1989). The t ime constant of deact ivat ion of the NMDA channels de te rmines how long this de- lay can be.

The d e p e n d e n c e of synaptic modif icat ion (Aw) on the t iming of pre- and postsynaptic activ- ity in the t a /gamma networks is i l lustrated in Fig- ure 4. W h e n m e m o r y 1 is activated, glutamate is re leased f rom the axons of the pyramidal cells cod- ing for m e m o r y 1. The glutamate b inds to the NMDA receptors on all postsynapt ic pyramidal cells and leads to an activated state (Fig. 4B1). Channels in the activated state can open, but only if the postsynapt ic cell depolar izes to rel ieve the Mg 2+ block. W h e n opening occurs, there is an influx of Ca 2+ (no t m o d e l e d expl ic i t ly) wh ich leads to LTP. This increases the synaptic s trength by a value (Awq) propor t ional to the convolut ion of the t ime course of the activated state of the NMDA channels and the postsynapt ic depolariza- t ion (see Material and Methods).

Figure 4 shows that the synaptic connec t ions that are s t reng thened in a the ta /gamma ne twork d e p e n d cri t ically on the t ime constant of deacti- vat ion of NMDA channels (unb ind ing of gluta- mate) . If the NMDA channels have fast deactiva- t ion and therefore span only one gamma cycle (Fig. 4B1), then cells that fire in different gamma cycles (Fig. 4A) and therefore represent different memor ies , wil l not have the synapses b e t w e e n them s t reng thened (Fig. 4B2, Awij ). The only con- nect ions that are s t rengthened are be tween cells that r epresen t the same memory. The synapses that are s t r eng thened are thus appropriate for au- toassociative LTM. If however , NMDA channels have a s low deact ivat ion that spans several gamma cycles (Fig. 4C1 ), connec t ions would be strength- ened b e t w e e n cells that represent different mem- ories (Fig. 4C2). This is what we call heteroasso- ciative memory . In this paper we wil l assume that

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Figure 4: Rules for LTP induction indicate that the type of memory formed in networks with theta/gamma oscil- lations will depend on the time constant of the NMDA channel. (A) Representation of the seven different mem- ories becoming active within a theta cycle. (B1) The time course of fast NMDA channels activated by mem- ory item one. (B2) Strong connections form only be- tween cells that fire in the same gamma cycle. Because these cells encoded the same memory, the fast NMDA channel forms the connections needed for autoassocia- tion. Awij represents the synaptic connections between cell i and j. (Cl) The time course of slow NMDA chan- nels whose time constant spans multiple gamma cycles. (C2) Connections form between memory item one and subsequent memories (i.e., heteroassociations are formed)

the ne twork contains only fast NMDA channels and is therefore special ized for the format ion of autoassociative LTM.

FORMATION OF LTM FOR NOVEL ITEMS

We now consider h o w a novel autoassociative m e m o r y can be incorpora ted into LTM in a phys- iologically realistic way in a ne twork wi th fast NMDA channels. This physiological real ism in- cludes the assumpt ion that synaptic modif ica t ion requires substantial repet i t ion (see Introduct ion, Materials and Methods). We also have added ran- dom m e m b r a n e noise not p resen t in our previous simulations (Figs. 2 and 3; Lisman and Idiart

1995). Our s imulat ion in Figure 5 shows the forma-

tion of LTM for mul t ip le i tems p resen ted in real

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FORMATION OF ASSOCIATIVE MEMORY

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t ime to the network. In the example, two novel m e m o r i e s are p re sen ted to the ne twork (arrows in Fig. 5A). The ADP leads to the firing of both mem- ories on subsequen t theta cycles, but wi th in a few cycles, the firing is progressively desynchronized. This is because the m e m b r a n e noise generates er- rors in the ADP t iming ramps and causes cells to fire in the w r o n g gamma subcycle. Because the ADP that controls t iming is unitary, there is no basis for error correct ion. Thus, once a t iming er- ror occurs for a cell, the error cont inues on sub- sequent theta cycles. As seen in Figure 5B, the repe t i t ion of memor i e s in STM gradually leads to the bui ld up of synaptic s t rength in LTM. The m e m o r y informat ion is thus learned. However, to

the ex tent that some cells par t ic ipat ing in a mem- ory representa t ion eventual ly fire in the wrong gamma cycle, errors wil l also b e c o m e synaptical ly encoded. This is i l lustrated by the inset in Figure 5B. The size of each matr ix e l e m e n t represents the strength of the synapse f rom cell i to cell j . We conclude that novel memor i e s can be incorpo- rated into LTM after a single presentat ion, bu t the encoding wil l inevi tably conta in inaccuracies.

Although a novel i tem cannot accurate ly be encoded into LTM by a single presentat ion, we thought that mul t ip le presenta t ions of the same sequence might lead to correc t encod ing if erro- neous information stored dur ing a previous pre- sentat ion could be r emoved dur ing a subsequent presenta t ion by LTD (for review, see Christ ie et al. 1994; Linden 1994; Debanne and Thompson 1996). We mode l LTD by decreas ing the synaptic conduc tance if a presynapt ic act ivat ion occurs wi thout postsynapt ic activation or vice versa. The t ime w i n d o w for w h i c h the pre- and postsynapt ic events are cons idered synchronous is d e t e r m i n e d by the t ime character is t ic of the NMDA channe l (see Materials and Methods) . Hence, if some syn- apses had been er roneous ly s t rengthened, one could expec t that re in t roduc t ion of the m e m o r y pat terns wil l reduce the e r roneous ly s t reng thened synapses. This is, however , not the case if the same sequence of m e m o r y pat terns is repea ted ly pre- sented. Some of the synapses that we re errone- ously s t rengthened dur ing initial presenta t ions do n o t get weakened (Fig. 6A).

Correct encoding can occur if the order of the same m e m o r y pat terns is varied f rom presenta t ion to presentat ion. We have appl ied this p rocedure in Figure 6B. Initially, some errors we re encoded into the synaptic matrix. After the sequence was presen ted repeatedly wi th the order in terchanged, a correct m e m o r y trace bui lds up.

D i s c u s s i o n

MECHANISM OF STM

It has been proposed that STM could rely on reverbatory mechan i sms ( H e b b 1949), and that the recur ren t connec t ions of attractor ne twork are an example of such a reverbatory loop (Amit 1989). Specifically, in attractor ne tworks activity is main ta ined because cells that r epresen t a mem- ory select ively exci te each other w i th strong syn-

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Figure 6: Conditions required for estab- lishing error-free autoassociative LTM of individual items. Memories are presented in rapid succession during a learning pe- riod. Such presentations are repeated many times and the resulting synaptic weight matrix is shown as a function of the number of presentations. The matrix ele- ments of 1-5 on the x- and y-axis repre- sents the synapses ideally coding for the first memory. The next memory is ideally coded by the elements at 6-10 at the x- and y-axis and so forth. (A) If the same eight memories are always introduced in the same order, the memories are never accurately learned, as shown by the inap- propriate components of the matrix. (B) If the eight memories are introduced in ran- dom order, the matrix becomes accurate. The simulations were done the following way: A sequence of items was selected from a eight-item list and introduced to the system. The sequence was kept active for 20 theta cycles allowing the individual memories to become encoded into the synapses. Then a new sequence was se- lected and introduced to the network. In A seven consecutive items was select from a list of eight elements and introduced to the network. In B the seven items were se- lected randomly from the list before each introduction. The final synaptic values ranged from 0 to 0.8.

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apses. By definition, this selectivity cannot exist for novel memories . Thus, if repeated firing in STM is necessary to produce LTM of novel memories , the firing cannot be produced by recurrent con- nections. This leaves intrinsic membrane changes ( such as the ADP) as the only known possibility for sustaining firing. This mechanism is nonredun- dant and thus cannot be error correcting. We con- c lude that STM for novel i tems is necessarily inac- curate. The degree of inaccuracy will depend on the detailed membrane properties and the level of cellular noise. We are unable to estimate h o w fast

STM will be disrupted by these factors. The fidelity of STM for known items is much greater, as wil l be described in the next paper (Jensen and Lisman 1996b, this issue).

REPETITION BY STM INCORPORATES NOVEL ITEMS INTO LTM

Our results show that an integrated STM and LTM network can capture multiple, novel memory items in real t ime and incorporate them

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into LTM. The STM mechanism produces a repe- tition of each item once every theta cycle and thereby leads to the gradual incorporation of each item into LTM (Fig. 5). This repetition is crucial because physiological analysis has shown that the synaptic modifications that underlie LTM occur slowly (Larson et al. 1986; Hanse and Gustafsson 1994).

To the extent that errors occur in STM be- cause of noise, as discussed above, the LTM that is slowly formed will be inaccurate (Fig. 5). There appears to be an interesting optimization problem here: The longer STM is maintained, the greater the encoding into LTM. However, the longer STM is maintained, the greater the chance for the cor- ruption of information by noise. We cannot deter- mine where the optima lies, but what is clear is that the encoding of autoassociative LTM will nec- essarily be inaccurate after a single presentation.

Figure 6 shows that accurate encoding can be achieved if multiple presentations are given. Each time the item is presented again, different errors will occur and some of the previous errors will be erased by activity-dependent depression of syn- apses (Fig. 6B). Eventually, a reasonably accurate LTM is formed by a kind of averaging process. One more generalization should be noted. If a novel memory item is always presented in the same tem- poral context (i.e., with the same items preceding and succeeding), erroneously encoded patterns tend to build up rather than get averaged out (Fig. 6A). To avoid this a novel memory item must ap- pear in different temporal context with each pre- sentation. This makes intuitive sense. If we are learning new words of a foreign language and a set of words always appears together in the same or- der, we will not be able to learn each word as an individual item. Only if the novel words appear separately, interleaved in different contexts, will we be able to extract them as single entities. Mc- Clelland et al. (1995) also argue for the impor- tance of interleaved learning.

ROLE FOR FAST NMDA CHANNELS

The study of NMDA channels has shown that different NMDA channels have different kinetics (Hestrin et al. 1992; Monyer et al. 1994; S. Vicini, J.F. Wang, J.S. Poling, and D.R. Grayson, pers. comm.) Recombinant NMDA receptors composed by the NMDARla subunit combined with the NMDAR2A subunit have a much faster deactiva-

tion time constant ( ~ 3 6 msec) than receptors with 2B or 2C subunits ( ~ 3 0 0 msec). These time constants are measured at room temperature and will be much shorter at body temperature given a Qlo of ~3 (Hestrin et al. 1990; Feldmeyer et al. 1993). NMDA channels with fast kinetics have been observed in cortical networks (Carmignoto and Vicini 1992; Monyer et al. 1994). The role of this diversity has been completely unclear from a computational point of view. This paper provides the first indication of the reason for this diversity and the particular function that fast NMDA chan- nels might perform. Fast NMDA channels have time constants that do not span multiple gamma cycles. They will thus lead to autoassociative memory formation in networks with theta/gamma oscillations. The reader is referred to the Discus- sion section of the next paper in this series (Jensen and Lisman 1996b, this issue) for a discussion of other biological aspects of this model.

Acknowledgments We thank Larry Abbott and Jean-Marc Fellous for

reading the manuscript. This work was supported by the W.M. Keck Foundation, a National Institutes of Health grant (NS 27337) and the Alfred P. Sloan Foundation (94-10-1).

The publication costs of this article were defrayed in part by payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 USC section 1734 solely to indicate this fact.

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Received June 25, 1996; accepted in revised form September 25, 1996.

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